Question

Let $A = \begin{pmatrix}1&0&0\\ 2&1&0\\ 3&2&1\end{pmatrix}$ If $u_1$ and $u_2$ are column matrices such that $Au_{1} = \begin{pmatrix}1\\ 0\\ 0\end{pmatrix}$ and $ Au_{2} = \begin{pmatrix}0\\ 1\\ 0\end{pmatrix},$ then $ u_{1} + u_{2} $ is equal to :

• A. $\begin{pmatrix}-1\\ 1 \\ 0\end{pmatrix}$
• B. $\begin{pmatrix}- 1\\ 1 \\ -1 \end{pmatrix}$
• C. $\begin{pmatrix} -1\\ -1 \\ 0\end{pmatrix}$
• D. $\begin{pmatrix}1\\ -1 \\ -1 \end{pmatrix}$

Answer 1

Answer: (D)

$A\left(u_{1}+u_{2}\right) = \begin{pmatrix}1\\ 1\\ 0\end{pmatrix}\quad\quad|A| = 1$
$A^{-1} = \frac{1}{\left|A\right|}adj A $
$u_{1} + u_{2} = A^{-1} \begin{bmatrix}1\\ 1\\ 0\end{bmatrix}\quad\quad A^{-1} = \begin{bmatrix}1&0&0\\ -2&1&0\\ 1&-2&1\end{bmatrix} = \begin{bmatrix}1\\ -1\\ -1\end{bmatrix}$